Relation Theory • 6

Posted on November 4, 2021 by Jon Awbrey

Relation Theory • Species of Dyadic Relations

Returning to 2‑adic relations, it is useful to describe several familiar classes of objects in terms of their local and numerical incidence properties.  Let L \subseteq S \times T be an arbitrary 2‑adic relation.  The following properties of L can be defined.

Dyadic Relations • Total • Tubular

If L \subseteq S \times T is tubular at S then L is called a partial function or a prefunction from S to T.  This is sometimes indicated by giving L an alternate name, for example, {}^{\backprime\backprime} p {}^{\prime\prime}, and writing L = p : S \rightharpoonup T.  Thus we have the following definition.

\begin{matrix} L & = & p : S \rightharpoonup T & \text{if and only if} & L & \text{is} & \text{tubular} & \text{at}~ S. \end{matrix}

If L is a prefunction p : S \rightharpoonup T which happens to be total at S, then L is called a function from S to T, indicated by writing L = f : S \to T.  To say a relation L \subseteq S \times T is totally tubular at S is to say it is 1-regular at S.  Thus, we may formalize the following definition.

\begin{matrix} L & = & f : S \to T & \text{if and only if} & L & \text{is} & 1\text{-regular} & \text{at}~ S. \end{matrix}

In the case of a function f : S \to T, we have the following additional definitions.

Dyadic Relations • Surjective, Injective, Bijective

Resources

cc: Category Theory • Cybernetics (1) (2) • Ontolog Forum (1) (2)
cc: Structural Modeling (1) (2) • Systems Science (1) (2)
cc: FB | Relation Theory • Laws of Form • Peirce List

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